Algebra Colloquium

In this paper, we compute the Frobenius dimension of any cluster-tilted algebra of finite type. Moreover, we give conditions on the bound quiver of a cluster-tilted algebra $\mathrm{\Lambda }$ such that $\mathrm{\Lambda }$ has non-trivial open Frobenius structures.

We study the algebraic structure of rings $R$ whose zero-divisor graph $\mathrm{\Gamma }(R)$has clique number four. Furthermore, we give complete characterizations of all the finite commutative local rings with clique number 4.

In this paper, we mainly give characterizations of EP elements in terms of equations. In addition, a related notion named a central EP element is defined and investigated. Finally, we focus on characterizations of a generalized EP element, i.e., $\ast$-DMP element.

Let $\sigma$ be an automorphism and$\delta$ be a $q$-skew $\sigma$-derivation of an $F$-algebra $A$. We prove that if $A$ is semiprimitive and $\delta$ is algebraic, then the subalgebra $A^{\delta }=\{r\in A | \delta (r)=0\}$ has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field $F$ is uncountable. Then we apply it to actions of the $n^2$-dimensional Taft Hopf algebra $T_{n^2}(q)$ and the $q$-analogue $U_q(s\ell (2))$ of the enveloping algebra of the Lie algebra $s\ell (2)$.

Let $R$ be a domain. In this paper, we show that if $R$ is one-dimensional, then $R$ is a Noetherian Warfield domain if and only if every maximal ideal of $R$ is 2-generated and for every maximal ideal$M$ of $R$, $M$ is divisorial in the ring $(M:M)$. We also prove that a Noetherian domain $R$ is a Noetherian Warfield domain if and only if for every maximal ideal $M$ of $R$, $M^2$ can be generated by two elements. Finally, we give a sufficient condition under which all ideals of $R$ are strongly Gorenstein projective.

We construct a level -1/2 vertex representation of the quantum $N$-toroidal algebra of type $C_n$, which is a natural generalization of the usual quantum toroidal algebra. The construction also provides a vertex representation of the quantum toroidal algebra for type $C_n$ as a by-product.

In this paper we define the notion of Brauer–Clifford group for $(S,\mathcal{L})$-Azumaya algebras when $S$ is a commutative algebra and$\mathcal{L}$ is a $(k,S)$-Lie algebra over a commutative ring $k$. This is the situation that arises in applications having connections to differential geometry. This Brauer–Clifford group turns out to be an example of a Brauer group of a symmetric monoidal category.

We present a deformation theory associated to the higher Hochschild cohomology $H_{S^2}^{\ast }(A,A)$. We also study a $G$-algebra structure associated to this deformation theory.

A Bihom-Lie algebra is a generalized Hom-Lie algebra endowed with two commuting multiplicative linear maps. In this paper, we study representations of Bihom-Lie algebras. In particular, derivations, central extensions, derivation extensions, the trivial representation and the adjoint representation of Bihom-Lie algebras are studied in detail.

We define perfect ideals, near perfect ideals and upper bounded ideals of a finite-dimensional Lie superalgebra, and study the properties of these three kinds of ideals through their relevant sequences. We prove that a Lie superalgebra is solvable if and only if its maximal perfect ideal is zero, or its quotient superalgebra by the maximal perfect ideal is solvable. We also show that a Lie superalgebra is nilpotent if and only if its maximal near perfect ideal is zero. Moreover, we prove that a nilpotent Lie superalgebra has only one upper bounded ideal, which is the nilpotent Lie superalgebra itself.

For a prime $p$and a positive integer$k$, let $q=p^k$ and ${\mathbb{F}}_{q^n}$ be the extension field of ${\mathbb{F}}_q$. We derive a sufficient condition for the existence of a primitive element $\alpha$ in${\mathbb{F}}_{q^n}$ such that ${\alpha }^3-\alpha +1$ is also a primitive element of ${\mathbb{F}}_{q^n}$, a sufficient condition for the existence of a primitive normal element $\alpha$ in ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$ such that ${\alpha }^3-\alpha +1$ is a primitive element of ${\mathbb{F}}_{q^n}$, and a sufficient condition for the existence of a primitive normal element $\alpha$ in ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$ such that ${\alpha }^3-\alpha +1$ is also a primitive normal element of ${\mathbb{F}}_{q^n}$ over ${\mathbb{F}}_q$.

Let $R$ be a commutative ring and $U(R)$ the multiplicative group of unit elements of $R$. In 2012, Khashyarmanesh et al. defined the generalized unit and unitary Cayley graph, $\Gamma (R,G,S)$, corresponding to a multiplicative subgroup $G$ of $U(R)$ and a nonempty subset $S$ of $G$ with $S^{-1}=\{s^{-1} | s\in S\}\subseteq S$, as the graph with vertex set $R$and two distinct vertices $x$ and $y$ being adjacent if and only if there exists $s\in S$ such that $x+sy\in G$. In this paper, we characterize all Artinian rings $R$ for which $\Gamma (R,U(R),S)$ is projective. This leads us to determine all Artinian rings whose unit graphs, unitary Cayley graphs and co-maximal graphs are projective. In addition, we prove that for an Artinian ring $R$ for which $\Gamma (R,U(R),S)$ has finite nonorientable genus, $R$ must be a finite ring. Finally, it is proved that for a given positive integer $k$, the number of finite rings $R$ for which $\Gamma (R,U(R),S)$ has nonorientable genus $k$ is finite.